A050940 -id:A050940 - cp's OEIS frontend

A034707 Numbers that are sums (of a nonempty sequence) of consecutive primes.

2, 3, 5, 7, 8, 10, 11, 12, 13, 15, 17, 18, 19, 23, 24, 26, 28, 29, 30, 31, 36, 37, 39, 41, 42, 43, 47, 48, 49, 52, 53, 56, 58, 59, 60, 61, 67, 68, 71, 72, 73, 75, 77, 78, 79, 83, 84, 88, 89, 90, 95, 97, 98, 100, 101, 102, 103, 107, 109, 112, 113, 119, 120, 121, 124, 127

Offset: 1

Erich Friedman

A050936 is a subsequence (which still includes primes, embodied by A067377). - Enoch Haga, Jun 16 2002, R. J. Mathar, Oct 10 2010

T. D. Noe, Table of n, a(n) for n = 1..10000
Leo Moser, On the Sum of Consecutive Primes. Canad. Math. Bull. 6 (1963), 159-161.
Janyarak Tongsomporn, Saeree Wananiyaku, and Jörn Steuding, Sums of consecutive prime squares, Integers (2022) Vol. 22, #A9.

Complement is A050940.

Cf. A050936, A054845.

f[n_] := Block[{len = PrimePi@ n}, p = Prime@ Range@ len; Count[ Flatten[ Table[ p[[i ;; j]], {i, len}, {j, i, len}],1], q_ /; Total@ q == n]]; Select[ Range@ 1000, f@ # > 0 &] (* Or quicker for a larger range *)
lmt = 10000; p = Prime@ Range@ PrimePi@ lmt; t = Table[0, {lmt}]; Do[s = 0; j = i; While[s = s + p[[j]]; s <= lmt, t[[s]]++; j++], {i, Length@ p}]; Select[ Range@ lmt, t[[#]] > 0 &]
upto=200;Select[Union[Flatten[Table[ Total/@Partition[Prime[ Range[ PrimePi[ upto]]],n,1],{n,upto-1}]]],#<=upto&] (* Harvey P. Dale, Jul 15 2011 *)

is(n)=if(isprime(n), return(1)); my(v,m=1,t); while(1, v=vector(m++); v[m\2]=precprime(n\m); for(i=m\2+1,m,v[i]=nextprime(v[i-1]+1)); forstep(i=m\2-1,1,-1,v[i]=precprime(v[i+1]-1)); if(v[1]==0, return(0)); t=vecsum(v); if (t==n, return(1)); if(t>n, while(t>n, t-=v[m]; v=concat(precprime(v[1]-1), v[1..m-1]); t+=v[1]), while(tCharles R Greathouse IV, May 05 2016

A054845(a(n)) > 0. - Ray Chandler, Sep 20 2023

Updated a misleading comment. - R. J. Mathar, Oct 10 2010

A037174 Primes which are not the sum of consecutive composite numbers.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 47, 61, 73, 107, 167, 179, 313, 347, 421, 479, 719, 863, 1153, 1213, 1283, 1307, 1523, 3467, 3733, 4007, 4621, 4787, 5087, 5113, 5413, 7523, 7703, 9817, 10333, 12347, 12539, 13381, 17027, 18553, 19717, 19813, 23399, 26003, 31873, 36097, 38833

Offset: 1

Naohiro Nomoto

nonn

It seems reasonable that a(n)/A079149(n) has an asymptote that could be estimated. - Peter Munn, Aug 21 2023

T. D. Noe, Table of n, a(n) for n = 1..3492 (terms less than 2*10^9)

Subsequence of A079149.
With {1}, the complement of A133576.
Primes that are the sum of specific numbers of consecutive composite numbers: A060254 (2), A060328 (3), A060329 (4), A060330 (5), A060331 (6), A060332 (7), A060333 (8).
Cf. A050940, A197227.

N:= 5000:
primes,comps:= selectremove(isprime,{$2..N}):
M:= nops(comps):
X:= primes:
for n from 1 to floor(sqrt(2*N)) do
i:= 1;
T:= add(comps[k],k=1..n);
while T <= N do
X := X minus {T};
if i + n > M then break fi;
T := T + comps[i+n] - comps[i];
i := i+1;
od;
od:
X;
# Robert Israel, Jun 24 2008

More terms from Jud McCranie, Jul 12 2000

Corrected by T. D. Noe, Aug 15 2008

A197227 Primes that are not the sum of at least two consecutive primes.

Original entry on oeis.org

2, 3, 7, 11, 13, 19, 29, 37, 43, 47, 61, 73, 79, 89, 103, 107, 113, 137, 149, 151, 157, 163, 167, 179, 191, 193, 227, 229, 239, 241, 257, 277, 283, 293, 307, 313, 317, 337, 347, 359, 367, 383, 389, 397, 409, 419, 433, 461, 467, 509, 521, 541, 547, 557, 569

Offset: 1

T. D. Noe, Nov 03 2011

nonn

Complement of A067377 in the primes. For the primes less than 10^6, these primes make up about 56%.

Robert Israel, Table of n, a(n) for n = 1..5000

Cf. A050940, A067377, A307610.

P:= [seq(ithprime(i),i=1..10^3)]:
S:= ListTools:-PartialSums([0,op(P)]):
sort(convert(convert(P,set) minus {seq(seq(S[i]-S[j],j=1..i-2),i=1..10^3+1)},list)); # Robert Israel, May 09 2021

lim = 1000; pFound = {}; ps = Prime[Range[PrimePi[lim]]]; sm = ps; i = 0; While[i++; j = 1; While[sm[[j]] = sm[[j]] + ps[[i + j]]; sm[[j]] <= lim, If[PrimeQ[sm[[j]]], AppendTo[pFound, sm[[j]]]]; j++]; j > 1]; Complement[ps, pFound]

Prime(n) such that A307610(n) = 1. - Ray Chandler, Sep 21 2023

Definition clarified by Jonathan Sondow, May 18 2013

A074192 Number of numbers 0 <= m < 10^n which are not the sum of one or more consecutive primes.

Original entry on oeis.org

5, 47, 520, 5191, 51462, 518001, 5205110, 52209546, 522955756

Offset: 1

Jean-christophe Colin (jc-colin(AT)wanadoo.fr), Sep 19 2002

If the upper limit is changed to include 10^n, the entries are increased by 1 for n = 3, 4, 5,.. and perhaps others, because 1000, 10000 and 100000 are not sums of consecutive primes, whereas 10=2+3+5 and 100=2+3+5+...+19+23 are. [R. J. Mathar, Oct 10 2010]
Based on work of Enoch Haga, Jud McCranie and Felice Russo.
The ratio a(n)/10^n might be converging to 0.52..., it might also be slowly growing to 1, which way is it? - Daniel Forgues, Nov 03 2011

C. Rivera, Puzzle 183

Cf. A050940.

Definition corrected by R. J. Mathar, Oct 10 2010

a(8) corrected and a(9) from Sean A. Irvine, Jan 14 2025

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A034707 Numbers that are sums (of a nonempty sequence) of consecutive primes.

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A037174 Primes which are not the sum of consecutive composite numbers.

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A197227 Primes that are not the sum of at least two consecutive primes.

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Maple

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A074192 Number of numbers 0 <= m < 10^n which are not the sum of one or more consecutive primes.

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